2. Key Parametric Curves in CAD

Hermite Curves

Defined by: Two endpoints and tangent vectors at each. Useful for: Controlled transitions, animation paths, and local modification. Equation: $C(t) = h_1(t)P_0 + h_2(t)P_1 + h_3(t)T_0 + h_4(t)T_1$ where $P_0, P_1$ are endpoints, $T_0, T_1$ are tangents, and $h_i(t)$ are basis functions.

Hermite curves are a type of parametric curve defined by two endpoints and the tangents at those points. This means that not only do we specify where the curve starts and ends, but we also control how the curve approaches each endpoint (its direction and steepness) using tangent vectors. The equation provided allows us to calculate points on the curve by using basis functions that depend on a parameter 't'. Adjusting the tangents allows for smooth transitions and flexible animation paths.

Imagine you are drawing a path on a piece of paper. You can specify where the path starts (the first endpoint) and where it finishes (the second endpoint). Now, if you also have control over how steep your path comes into those points, you can create a curve that smoothly transitions, just like how a car approaches and leaves a curved road.

Bézier Curves

Defined by: A set of $n+1$ control points $P_0, ... , P_n$ . Equation: $B(t) = \\sum_{i=0}^{n} {n \choose i} (1-t)^{n-i} t^i P_i$, $t \in [0, 1]$ Key Properties:

• Curve starts at $P_0$ and ends at $P_n$.
• Entire curve lies within the convex hull of the control points.
• Widely used in graphic design, CAD, and animation.

Bézier curves use a series of control points to define the shape of the curve. The curve is mathematically constructed so that it starts at the first control point and ends at the last one, always remaining inside the shape formed by connecting all the control points (this shape is called the convex hull). The equation demonstrates how each point on the curve is a blend of the control points, weighted by a parameter 't' that ranges from 0 to 1. This makes Bézier curves particularly useful in graphic design and CAD applications.

Think of Bézier curves like guiding a boat along a series of buoys in a lake. The buoys represent the control points. The path of the boat (the curve) must start at the first buoy and end at the last one, traveling smoothly within the limits defined by all the buoys around it.

B-spline Curves

Generalization of Bézier curves enabling local shape control. Defined by: Control points, degree $p$, and knot vector (partition of the parameter domain). Equation: $C(t) = \sum_{i=0}^{n} N_{i,p}(t) P_i$ where $N_{i,p}(t)$ are B-spline basis functions. Properties:

• Modifying a control point affects only a segment (locality).
• Versatile degree and smoothness adjustment.

B-spline curves are a more flexible and powerful extension of Bézier curves. While they still use control points to determine the shape, they allow for more localized adjustments; changing one control point will only affect a small portion of the curve, not the entire shape. This is because B-splines are defined by both a parameter called 'degree' (which controls the smoothness of the curve) and a 'knot vector' (which helps partition the parameter space). The equation illustrates how the position on the curve is determined by these basis functions, weighted by control points.

Imagine stretching a rubber band over a series of pegs. If you adjust one peg (the control point), the rubber band only changes slightly in that area, leaving the rest of the shape intact. This is similar to how a B-spline allows for local control over curves.

Rational Curves - NURBS

Non-Uniform Rational B-Splines - NURBS. Most general form, includes weights for exact representation of conic sections. Equation: $C(t) = \\frac{\sum_{i=0}^{n} N_{i,p}(t) w_i P_i}{\sum_{i=0}^{n} N_{i,p}(t) w_i}$ Used for: Circles, ellipses, and flexible freeform curves in modern CAD.

NURBS, or Non-Uniform Rational B-Splines, are the most versatile type of curve in CAD because they allow for the inclusion of weights, which help accurately represent complex shapes such as circles and ellipses. The equation presented shows how the curve is generated by pulling control points in various directions based on their assigned weights, resulting in more precision. This makes NURBS especially powerful for designing freeform shapes, which are essential in modern CAD applications.

Consider NURBS like adjusting a group of balloons with strings attached to different weights. If you pull on a string (a control point) with a heavier weight, it influences the overall shape more than if you pulled on a lighter one. This weight adjustment allows for precise control over how the balloons (points) form a particular shape.

--